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%% section* Appendix C [slac-pub-7204-0-0-5u3 in slac-pub-7204-0-0-5u3: slac-pub-7204-0-0-5u4]
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\section*{\usemenu{slac-pub-7204::context::slac-pub-7204-0-0-5u3}{Appendix C}}\label{section::slac-pub-7204-0-0-5u3}
Here we give some technical details of the calculation leading to the
diffractive contributions of Sections \docLink{slac-pub-7204-0-0-4.tcx}[lcs]{4.3} and \docLink{slac-pub-7204-0-0-4.tcx}[tcs]{4.4}.
The main problem is the calculation of the partonic part ${\cal M}$ of the
amplitude given by Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[saab]{39}). In the following, the necessary
manipulations will be described for the longitudinal cross section in the
case of high-$p_\perp$ quark-antiquark jets. A complete derivation of
$F_L^{D,1}$, Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[fldi]{53}), is obtained. For the other diffractive
contributions only the results for the corresponding partonic parts
${\cal M}$ will be given.
The longitudinal cross section can be calculated using the relation
$\sigma_L=(Q^2/q_0^2)\sigma_{00}$. Therefore it is sufficient to calculate
${\cal M}$, given by Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[mcal]{40}), for the unphysical photon
polarization $\epsilon(q)=(1,\vec{0})$,
\beq
{\cal M}_0=\bar{u}_{s'}(\bar{p})\left[\gamma_0\frac{\qs-\bps}{(q-\bar{p})^2}
\epsilons-\epsilons\frac{\qs-\bls}{(q-\bar{l})^2}\gamma_0\right]v_{r'}
(\bar{l})\, .\label{ma1}
\eeq
Here $\epsilon_{\lambda'}(k)=\epsilon$ has been used for brevity. The
high-energy approximation for the quark propagator introduced in
Sect.~\docLink{slac-pub-7204-0-0-3.tcx}[qq]{3} (see in particular Eqs. (\docLink{slac-pub-7204-0-0-3.tcx}[prop1]{15}),(\docLink{slac-pub-7204-0-0-3.tcx}[prop2]{16}))
corresponds to the replacements
\bea
\qs-\bps&\simeq&\bls+\ks=\sum_{\rho}v_\rho(\bar{l})\bar{v}_\rho(\bar{l})+\ks
\\ \nonumber\\ \qs-\bls&\simeq&\bps+\ks=\sum_{\rho}u_\rho(\bar{p})
\bar{u}_\rho(\bar{p})+\ks\, .
\eea
Note, that $k$ is an on-shell vector. Furthermore, it can be shown that the
terms proportional to $\ks$ are suppressed in Eq.~(\docLink{slac-pub-7204-0-0-5u3.tcx}[ma1]{105}) in the limit
$k_\perp^2\ll p_\perp^2\simeq l_\perp^2$, $\alpha' \ll 1$, which corresponds
to the case of high-$p_\perp$ $q\bar{q}$-jets. Making use of the relations
\beq
\bar{u}_{s'}(\bar{p})\epsilons u_\rho(\bar{p})=(2\bar{p}\epsilon)
\delta_{s'\rho}\quad,\quad\bar{v}_\rho(\bar{l})\epsilons v_{r'}(\bar{l})=
(2\bar{l}\epsilon)\delta_{\rho r'}
\eeq
the following expression for ${\cal M}_0$ can be derived,
\beq
{\cal M}_{0,q\bar{q}}=2\bar{u}_{s'}(\bar{p})\gamma_0v_{r'}(\bar{l})\left(
\frac{\bar{l}\epsilon}{(q-\bar{p})^2}-\frac{\bar{p}\epsilon}{(q-\bar{l})^2}
\right)\, .\label{ma2}
\eeq
Here the index $q\bar{q}$ specifies the considered kinematical region of
high-$p_\perp$ quark and antiquark.
With $\epsilon_-=2\epsilon_\perp k_\perp/k_+$, which follows from
$\epsilon k=0$, the relations
\beq
\bar{l}\epsilon=\frac{\alpha}{\alpha'}\epsilon_\perp k_\perp
-\epsilon_\perp l_\perp
\quad,\quad \bar{p}\epsilon=\frac{1-\alpha}{\alpha'}\epsilon_\perp k_\perp-
\epsilon_\perp p_\perp\label{lpe}
\eeq
are obtained. The denominators in Eq.~(\docLink{slac-pub-7204-0-0-5u3.tcx}[ma2]{109}) can be given in the form
\beq
(q-\bar{p})^2\simeq-\alpha N^2\quad,\quad (q-\bar{l})^2\simeq-(1-\alpha)N^2
\left(1+\frac{2k_\perp p_\perp}{\alpha(1-\alpha)N^2}+{\cal O}(k_\perp^2)
\right)\, ,\label{den}
\eeq
where
\beq
N^2=Q^2+\frac{p_\perp^2}{\alpha(1-\alpha)}\, .
\eeq
Although the term $\sim k_\perp$ in Eq.~(\docLink{slac-pub-7204-0-0-5u3.tcx}[den]{111}) is small compared to the
leading term, it can not be neglected. In combination with the term
$\sim k_\perp/\alpha'$ of Eq.~(\docLink{slac-pub-7204-0-0-5u3.tcx}[lpe]{110}) it will give a finite contribution
of order $k_\perp^2/\alpha'$.
Inserting (\docLink{slac-pub-7204-0-0-5u3.tcx}[lpe]{110}) and (\docLink{slac-pub-7204-0-0-5u3.tcx}[den]{111}) into Eq.~(\docLink{slac-pub-7204-0-0-5u3.tcx}[ma2]{109}) and keeping only
the leading contributions the following formula for ${\cal M}_{0,q\bar{q}}$
is obtained,
\beq
{\cal M}_{0,q\bar{q}}=\frac{2\bar{u}_{s'}(\bar{p})\gamma_0v_{r'}(\bar{l})}{
\alpha(1-\alpha)N^2}\left(\epsilon_\perp p_\perp+\frac{(2\epsilon_\perp
k_\perp)(p_\perp k_\perp)}{\alpha'N^2}\right)\, .
\eeq
Notice, that ${\cal M}_{0,q\bar{q}}$ depends on the intermediate gluon
momentum $k$ and
the integration over this variable has to be performed independently for
the amplitude and its complex conjugate. Therefore, we do actually not need
the square of ${\cal M}_{0,q\bar{q}}$ but the product
${\cal M}_{0,q\bar{q}}^*(k){\cal M}_{0,q\bar{q}}(\tilde{k})$. Here $k$ and
$\tilde{k}$ are two independent integration
variables. Summing over transverse gluon polarizations and quark and
antiquark helicities the following result is derived,
\beq
\sum_{\lambda's'r'}{\cal M}_{0,q\bar{q}}^*(k){\cal M}_{0,q\bar{q}}(\tilde{k})
=\frac{8q_+^2}{\alpha
(1-\alpha)N^4}\left(p_\perp+k_\perp\frac{(2p_\perp k_\perp)}{\alpha'N^2}
\right)\left(p_\perp+\tilde{k}_\perp\frac{(2p_\perp \tilde{k}_\perp)}{\alpha'
N^2}\right)\, .
\eeq
Using this expression and the colour factor of Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[cf1]{49}) together with
the amplitude (\docLink{slac-pub-7204-0-0-4.tcx}[saab]{39}) and the phase space formula (\docLink{slac-pub-7204-0-0-4.tcx}[dphi]{52}) the
cross section $\sigma_L$ can be calculated. Note, that in Eqs.~(\docLink{slac-pub-7204-0-0-4.tcx}[g1]{54}),
(\docLink{slac-pub-7204-0-0-4.tcx}[g2]{55}) the phase space integration over $\alpha'=\gamma'$ has been
substituted by an integration over $u$, defined in Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[udef]{57}).
For the other possible kinematical configurations in both the longitudinal
and the transverse case similar techniques can be used to perform the
calculation of ${\cal M}$. In the following we simply state the results for
the polarization sums over ${\cal M}^*(k){\cal M}(\tilde{k})$. The
kinematical variables are the same as in Sect.~\docLink{slac-pub-7204-0-0-4.tcx}[qqg]{4}.
For the longitudinal cross section in the case of high-$p_\perp$ quark and
gluon jets the following polarization sum is required,
\beq
\sum_{\lambda's'r'}{\cal M}^*_{0,qg}(l){\cal M}_{0,qg}(\tilde{l})=
\frac{4(1-\alpha)q_+^2}{N^4}\frac{l_\perp\tilde{l}_\perp}{\alpha''}\, .
\eeq
To derive $F_L^{D,2}$, Eq.~(\docLink{slac-pub-7204-0-0-4.tcx}[fldi]{53}), the contribution from
high-$p_\perp$ antiquark-gluon jets has to be added, which is proportional
to $\alpha$ and renders the cross section independent of the momentum
fraction.
The diffractive contribution to the transverse structure function from the
region of high-$p_\perp$ quark and antiquark jets can be calculated from
\bea
\frac{1}{2}\sum_{i=1,2}\sum_{\lambda's'r'}{\cal M}^*_{i,q\bar{q}}(k)
{\cal M}_{i,q\bar{q}}(\tilde{k}) = \frac{8(\alpha^2+(1-\alpha)^2)}{\alpha(1-
\alpha)N^4} \, \times
\eea
\bea
\left(\!\frac{p_\perp^ip_\perp^j}{\!\alpha(\!1\!-\!\alpha\!)}\!-\!\frac{k_\perp^i
k_\perp^j}{\alpha'}\!+\!\frac{(2p_\perp k_\perp)p_\perp^ik_\perp^j}{\!\alpha(\!1\!-\!\alpha\!)N^2\alpha'}\!-\!\frac{N^2}{2}\!\delta^{ij}\!\right)\!
\left(\!\frac{p_\perp^ip_\perp^j}{\alpha(\!1\!-\!\alpha\!)}\!-\!\frac{\tilde{k}_\perp^i\tilde{k}_\perp^j}{\alpha'}\!+\!\frac{(2p_\perp \tilde{k}_\perp)p_\perp^i\tilde{k}_\perp^j}{\alpha(\!1-\!\alpha\!)N^2\alpha'}\!
-\!\frac{N^2}{2}\!\delta^{ij}\!\right) . \nonumber
\eea
In the case of high-$p_\perp$ quark and gluon the corresponding contribution
reads
\bea
\frac{1}{2}\!\sum_{i=1,2}\!\sum_{\,\,\lambda's'r'}\!\!{\cal M}^*_{i,qg}(l)
{\cal M}_{i,qg}(\tilde{l})\! &=&\!\frac{4p_\perp^2(l_\perp\tilde{l}_\perp)}
{\alpha(\!1\!-\!\!\alpha)N^4\alpha''}\!\! \, \times \\
& & \!\!\left[\alpha\!+\!\frac{1}{\alpha}
\!\left(\!\frac{\alpha(\!1\!\!-\!\!\alpha)N^2}{p_\perp^2}\right)^{\!\!2}
\!\!\!+\!\frac{(\!1\!\!-\!\!\alpha)^2}{\alpha}\!\left(\!1\!-\!
\frac{\alpha(\!1\!\!-\!\!\alpha)N^2}{p_\perp^2}\right)^{\!\!2}\right]
\!. \nonumber
\eea
Together with the formulae of Sections \docLink{slac-pub-7204-0-0-4.tcx}[ampl]{4.1} and \docLink{slac-pub-7204-0-0-4.tcx}[cols]{4.2} the
remaining results of Sections \docLink{slac-pub-7204-0-0-4.tcx}[lcs]{4.3} and \docLink{slac-pub-7204-0-0-4.tcx}[tcs]{4.4} follow in a
straightforward manner.
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